Two dots race the same path, one at constant speed. The difference between them is what the arclength parameter fixes.
The ticks are equally spaced in t, not in distance. Where they bunch up, the curve is being traced slowly.
Add segments and watch the polyline close on the true length.
A parametrization tells you two different things at once: which points make up the curve, and how fast you travel through them. The first is the curve's own business. The second is entirely up to whoever wrote the formula, and it usually varies wildly along the way.
The tick marks make that visible. They're equally spaced in the parameter t, and they're plainly not equally spaced along the curve. Where they crowd together the point is dawdling; where they stretch apart it's racing. The speed |r′(t)| in the graph below is exactly that stretching factor.
Which is why length is ∫|r′(t)| dt and not just the change in t. You're adding up distance actually covered, and the speed is the conversion rate between parameter and distance at each instant. Stretch a chain of straight segments along the curve and let their number grow, and that's the same sum arriving from the other direction.
Reparametrizing by arclength is the act of throwing the arbitrary part away: re-run the curve so speed is exactly 1 everywhere, and the parameter becomes distance travelled. Race the two dots and watch one of them keep a steady pace while the other lurches. They trace precisely the same curve.